I'm pretty sure the Tascam 424mkIII would fit your needs, as described.
They're very reasonable for $329/new, over the counter and online, of which you might find a free shipping deal, too.
This stays in line with your budget, and you might have enough scratch left to score an SM57, cable & stand. Shop the big chain gear sales, though. A couple years ago, GC was blowing out SM57's w/cable and stand, for $80. That was a sale item, but there are many mic/cable/stand deals out there, if you shop hard enough.
People love recording on their computers, though, and it gives a whole other world of possibilities with your music, if you consider overall track count, efx and editing features. I'd say that a majority of people prefer recording onto their computers, IMO.
But, [IMO], recording on the computer is a very mousey, gadgety, & finicky platform to record on, sometimes, and even those made-for-DAW controller "mixer" interfaces leave a lot to be desired.
You have to be very much in love with [fond of] your computer, to really love recording on your computer. [MO]
Some people just flat-out don't like recording on the 'puter, and don't want to do it. Count me as one.
I think some "standalones", all-in-one-type recorder/mixer combos are often a better digital solution, but those units often have glaring inadequacies, such as track time limitations, track-at-once limitations, format or backup limitations. Shop the digital standalone recorder/mixer workstations very carefully. There's dozens of newer and older digital "porta's", and I could not address the features and limitations of them all. Let's just say, for the record, that I have the analog mixer, digital HD recorder, Fostex FD-4 & FD-8, plus, the one-of-a-kind Tascam 564 Minidisc Portastudio. Those are the 3 digi-porta's that I can endorse, but again, that's MO. YMMV.
In new, affordable 8-track all-in-one-er's, there's the Fostex MR-8, which at present time is HUGELY popular. It gives you a lot, for $299. It's another solid-state multitracker, that records up to 8 tracks, 2 at a time, and interfaces to the 'puter for it's up/downloads by USB. It's a very small, handy, self-contained unit, that has the typical 2-in mixer that these budget units tend to have. It records in cd standard digital format, FWIW. It's very popular unit, for pricepoint alone. 8-tracks/$299/new.
There are a lot of digital-memory 4-trackers out there now, too, but they're integrally linked with the 'puter for upload/download. Even though they're small and convenient, [IMO] they have some mixer inadequacies, [the 2-input mixer] and other issues [2-simul-max]. I don't like any of the solid-state 4-trackers out there very much, so I'm not inclined to recommend them. Some people like these digital-memory, solid-state, ultra-miniaturized 4-trackers a lot, and swear by them, but YMMV.
For all these solid-state, memory-stick based multitrack recorders, referenced above, if you want longer runtime, the larger memory cards will cost you extra money.
[IMO] To learn the art of production, mixing and recording, from the very ground up, that's not as gadgety in it's functions, but records and mixes well,... and you think you could function with 4-tracks,... you're best set with the 424mkIII, for new equipment.
Or, if you would consider used equipment, for less money than the 424mkIII/new, you could find a used Tascam 246, which is the 2nd gen vintage analog Tascam Portastudio, and [IMO] is one of the best in features. The 246 has a higher mixer function capacity than the 424mkIII, and many other Porta's. You'd have to shop Ebay wisely, to score the most low use and mint condition 246 as possible.
A mint or near-mint 246 can often be bought, used, for about $225. 15 years ago, as new, the MSRP of the 246 was $1350. It was very advanced for it's time, and is so feature-packed, that it stands up very well as compared to most other 4-trackers. The Tascam 246 is a great item to find as used/mint.
Check Ebay/Tascam to see what's on the used market. If budget is a huge concern, there's some fantastic bang-for-your-buck in Ebay deals, but buyer beware! PS: Fostex too. See: Ebay/Fostex.
I'm into genuine Tascam Portastudios, and I think they're a great place to start recording, especially for young people, recording starters, and people on a budget who want mixing options, good sound & affordable convenience. They're simple, easy to learn, & useful. They're also full-functioned enough to sustain viable recording options for more experienced users and more complex productions.
Pls don't ever let me catch you calling any non-Tascam multitracker a "Portastudio". Tascam is the Portastudio, and all the others are 'porta-studio-like-devices', porta's, standalones, DAW's, whatever, but not Portastudios.
[IMO] You get more "hands-on" orientation with a Portastudio, and you get more messages, menus, hangups, interference and eye strain in 'puter recording. The 'puter side of recording offers a lot, but it's not for everyone, so think about it, and keep your options open when shopping.
It's a question for which the answer could go on forever,... and nearly did!
People have alternate opinions. Those are mine. Have fun gear shopping and recording!
